More specifically, I defined the operations multiplication (non-Abelian, composition of permutations) and addition (Abelian, modulo addition of bounded naturals) over the same set, that can be seen as a bounded set of naturals or permutations equivalently. Is it formally correct to call it an algebra?
Is it acceptable to call it an algebra in any informal sense?
The two operations are not (at least doesn't seem to be) related to each other in any way like the usual ones (+ and *). Although, it would be interesting to know about an incidental relation.
Some entries like this seem to relate the concept of algebra to vectors, which is not my case (I suppose).
A use that seems non-related to vectors is in relational algebra which borrows the word algebra from the fact it uses algebraic structures.
Any set of (finitary) operations, obeying any list of axioms, can be called an algebra according to the terminology of the field of universal algebra. These are special cases of algebras for monads.
But generally the more interesting question is whether that class of operations and axioms is of a particular form, has particular behavior, etc. For example, my first question about your two operations is whether they obey a compatibility condition and what form it takes.